My wondering isn't idle, it's rampant.
The Platonic solids were done with interpenetrated cards (to show off a bit?). If each had a small slit in each corner and they were glued or spot welded along the edges, that would produce a continuous surface. In
lieu of calculations, I can just pick dimensions off the 3D model in the picture. Any two points are measurable.
Rather than fiberglass, highly-tempered aluminum sheet that would never form a compound curve would be my choice. For fiberglass, I'd go inflatable.